Determination
The Smith set can be calculated with the Floyd–Warshall algorithm in time Θ(''n''3) or Kosaraju's algorithm in time Θ(''n''2).Example
When there is a Condorcet winner—a candidate that is majority-preferred over all other candidates—the Smith set consists of only that candidate. Here is an example in which there is no Condorcet winner: There are four candidates: A, B, C and D. 40% of the voters rank D>A>B>C. 35% of the voters rank B>C>A>D. 25% of the voters rank C>A>B>D. The Smith set is . All three candidates in the Smith set are majority-preferred over D (since 60% rank each of them over D). The Smith set is not because the definition calls for the ''smallest'' subset that meets the other conditions. The Smith set is not because B is not majority-preferred over A; 65% rank A over B. (Etc.) In this example, under minimax, A and D tie; under Smith/Minimax, A wins. In the example above, the three candidates in the Smith set are in a "rock/paper/scissors" ''majority cycle'': A is ranked over B by a 65% majority, B is ranked over C by a 75% majority, and C is ranked over A by a 60% majority.Other criteria
Any election method that complies with the Smith criterion also complies with the Condorcet criterion, since if there is a Condorcet winner, then it is the only candidate in the Smith set. Obviously, this means that failing the Condorcet criterion automatically implies the non-compliance with the Smith criterion as well. Additionally, such sets comply with the Condorcet loser criterion. This is notable, because even some Condorcet methods do not (Minimax). It also implies the mutual majority criterion, since the Smith set is a subset of the MMC set.http://dss.in.tum.de/files/brandt-research/dodgson.pdf The Smith set and Schwartz set are sometimes confused in the literature. Miller (1977, p. 775) lists GOCHA as an alternate name for the Smith set, but it actually refers to the Schwartz set. The Schwartz set is actually a subset of the Smith set (and equal to it if there are no pairwise ties between members of the Smith set).Complying methods
The Smith criterion is satisfied by Ranked Pairs, Schulze's method, Nanson's method, the Robert's Rules method for voting on motions & amendments, and several other methods. Methods failing the Condorcet criterion also fail the Smith criterion. Some Condorcet methods, such asExamples
Minimax
:'' Mutual majority criterion#Minimax'' The Smith criterion implies the Mutual majority criterion, therefore Minimax's failure to satisfy the Mutual majority criterion is also a failure to satisfy the Smith criterion. Observe that the set S = in the example is the Smith set and D is the Minimax winner.See also
* Independence of Smith-dominated alternativesReferences